Equation 11.5 is the fundamental equation describing natural selection for a measured genotype at a single locus. Given that many adaptive traits are polygenic but with the underlying loci being unknown, Fisher (1930) elucidated many properties of natural selection using the theory of the quantitative genetics of unmeasured genotypes (Chapter 9). His central result is summarized in an equation that is the unmeasured genotype analogue of the measured genotype equation 11.5. Fisher called this equation the fundamental theorem of natural selection, which describes how natural selection operates upon the phenotype of fitness when fitness is regarded as a heritable but genetically unmeasured trait. There are many ways of deriving this theorem, and we will present only one of the simplest.
Because we assume that no genotypes are being measured, we must focus exclusively upon phenotypes. This means that we can no longer use the primary definition of fitness in population genetics as a genotypic value for a measured genotype; rather, we now use the evolutionary ecology definition of fitness as a measure of average reproductive success of a phenotypic class of individuals rather than a genotypic class. Therefore, let x be the phenotypic value of some trait for an individual in a population (which we assume is a continuous, quantitative trait), and let f (x) be the probability distribution that describes the frequencies of x in the population. The mean phenotype is then where the integration is over all possible values of x. We now assign a fitness value, w(x), to those individuals sharing a common phenotypic value x. The mean or average fitness of the population is
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